Lévy Distribution
X ~ Levy(μ, c)
Heavy-Tailed / Stable Family • Published July 22, 2026
The most extreme of the three stable distributions with elementary densities (alongside the already-published Normal and Cauchy): a full step beyond the Cauchy's undefined mean, since neither the mean nor the variance exists for any parameter value, verified via the exact reciprocal-of-squared-Normal construction (X = μ + c/Z²) confirmed by a Kolmogorov–Smirnov test against direct sampling, a closed-form median and characteristic function, a closed-form maximum-likelihood estimator for the scale given a known location, and a running-sample-mean demonstration of the divergence.
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About This Reference Sheet
The most extreme of the three stable distributions with elementary densities (alongside the already-published Normal and Cauchy): a full step beyond the Cauchy's undefined mean, since neither the mean nor the variance exists for any parameter value, verified via the exact reciprocal-of-squared-Normal construction (X = μ + c/Z²) confirmed by a Kolmogorov–Smirnov test against direct sampling, a closed-form median and characteristic function, a closed-form maximum-likelihood estimator for the scale given a known location, and a running-sample-mean demonstration of the divergence.
Support
x ∈ (μ, ∞)
Parameters
μ ∈ ℝ (location), c > 0 (scale)